A322403 -id:A322403 - cp's OEIS frontend

A322149 In the binary representation of n, replace each run of k 0's (or 1's) with k^2 0's (or 1's).

0, 1, 2, 15, 16, 5, 30, 511, 512, 33, 10, 47, 240, 61, 1022, 65535, 65536, 1025, 66, 271, 80, 21, 94, 1535, 7680, 481, 122, 495, 8176, 2045, 131070, 33554431, 33554432, 131073, 2050, 8207, 528, 133, 542, 8703, 2560, 161, 42, 175, 752, 189, 3070, 196607, 983040

Offset: 0

Rémy Sigrist, Nov 28 2018

This sequence has similarities with A001196: here we square the length of each run of consecutive equal bits, there we double it.

Rémy Sigrist, Table of n, a(n) for n = 0..16383
Index entries for sequences related to binary expansion of n

Cf. A000975, A001196, A005811, A322403.

squareList[v_] := Flatten[ConstantArray[v, {Length[v]}]]; a[n_] := FromDigits[ Flatten[squareList /@ Split[IntegerDigits[n, 2]]], 2]; Array[a, 60, 0] (* Amiram Eldar, Dec 07 2018*)

a(n) = if (n==0, 0, my (b=n%2, k=valuation(n+b,2)); (a(n\2^k) + b) * 2^(k^2) - b)

a(n) = A322403(n, n).
a(n) >= n with equality iff n belongs to A000975.
a(2^n) = 2^(n^2) for any n >= 0.
a(2^n - 1) = 2^(n^2) - 1 for any n >= 0.
A005811(a(n)) = A005811(n).

A322404 Square array T(n, k) read by antidiagonals, n >= 0 and k >= 0: the lengths of runs in binary expansion of T(n, k) are obtained by adding those of n and of k (see Comments for precise definition).

Original entry on oeis.org

0, 1, 1, 2, 3, 2, 3, 12, 12, 3, 4, 7, 12, 7, 4, 5, 24, 56, 56, 24, 5, 6, 51, 24, 15, 24, 51, 6, 7, 28, 3276, 112, 112, 3276, 28, 7, 8, 15, 28, 455, 48, 455, 28, 15, 8, 9, 48, 240, 120, 25368, 25368, 120, 240, 48, 9, 10, 99, 48, 31, 56, 51, 56, 31, 48, 99, 10

Offset: 0

Rémy Sigrist, Dec 06 2018

For any n >= 0 and k >= 0:
- let r_n be the lengths of runs in binary expansion of n,
- for n = 0: we assume that r_0 = (0),
- let R_n be the #r_n-periodic sequence whose first #r_n terms match r_n,
- r_{T(n, k)} has lcm(#r_n, #r_k) terms and r_{T(n, k)}(i) = R_n(i) + R_k(i) for i = 1..lcm(#r_n, #r_k).

			Array T(n, k) begins (in decimal):
  n\k|  0   1     2    3      4       5      6     7       8       9        10
  ---+------------------------------------------------------------------------
    0|  0   1     2    3      4       5      6     7       8       9        10
    1|  1   3    12    7     24      51     28    15      48      99       204
    2|  2  12    12   56     24    3276     28   240      48   12700       204
    3|  3   7    56   15    112     455    120    31     224     903      3640
    4|  4  24    24  112     48   25368     56   480      96   99896       792
    5|  5  51  3276  455  25368      51  29596  3855  199728      99  13421772
    6|  6  28    28  120     56   29596     60   496     112  116540       924
    7|  7  15   240   31    480    3855    496    63     960    7695     61680
    8|  8  48    48  224     96  199728    112   960     192  792688      3120
Array T(n, k) begins (in binary):
   n\k|     0       1            10         11              100
  ----+--------------------------------------------------------
     0|     0       1            10         11              100
     1|     1      11          1100        111            11000
    10|    10    1100          1100     111000            11000
    11|    11     111        111000       1111          1110000
   100|   100   11000         11000    1110000           110000
   101|   101  110011  110011001100  111000111  110001100011000
   110|   110   11100         11100    1111000           111000
   111|   111    1111      11110000      11111        111100000
  1000|  1000  110000        110000   11100000          1100000

See A322403 for the multiplicative variant.

Cf. A001196, A005811, A175046.

T(n,k) = my (v=0, p=1, rn=n, rk=k, b=if ((max(n,1)%2)&&(max(k,1)%2), 1, 0)); while (1, my (vn=if (rn==0, 0, valuation(rn+(rn%2), 2)), vk=if(rk==0, 0, valuation(rk+(rk%2), 2)), w=vn+vk); v+=b*p*(2^w-1); rn\=2^vn; rk\=2^vk; if (rn==0 && rk==0, return (v), rn==0, rn=n, rk==0, rk=k); p*=2^w; b=1-b)

For any m >= 0, n >= 0 and k >= 0:
- T(n, k) = T(k, n) (T is commutative),
- T(m, T(n, k)) = T(T(m, n), k) (T is associative),
- T(n, 0) = n (0 is a neutral element for T),
- T(n, 1) = A175046(n),
- T(n, n) = A001196(n),
- A005811(T(n, k)) = max(A005811(n), A005811(k), lcm(A005811(n), A005811(k))),
- T(2^n - 1, 2^k - 1) = 2^(n+k) - 1,
- T(2^n, 2^k) = 3 * 2^(n+k) when n > 0 and k > 0,
- T(n, k) is odd iff both n and k are odd.

cp's OEIS Frontend

A322149 In the binary representation of n, replace each run of k 0's (or 1's) with k^2 0's (or 1's).

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